| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 4.80·7-s + 8-s + 9-s − 0.882·11-s + 12-s − 2.82·13-s − 4.80·14-s + 16-s + 1.65·17-s + 18-s + 5.37·19-s − 4.80·21-s − 0.882·22-s + 5.85·23-s + 24-s − 2.82·26-s + 27-s − 4.80·28-s + 3.57·29-s + 10.4·31-s + 32-s − 0.882·33-s + 1.65·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.81·7-s + 0.353·8-s + 0.333·9-s − 0.266·11-s + 0.288·12-s − 0.783·13-s − 1.28·14-s + 0.250·16-s + 0.402·17-s + 0.235·18-s + 1.23·19-s − 1.04·21-s − 0.188·22-s + 1.22·23-s + 0.204·24-s − 0.554·26-s + 0.192·27-s − 0.908·28-s + 0.664·29-s + 1.87·31-s + 0.176·32-s − 0.153·33-s + 0.284·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.038567636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.038567636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.80T + 7T^{2} \) |
| 11 | \( 1 + 0.882T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 - 5.85T + 23T^{2} \) |
| 29 | \( 1 - 3.57T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 1.74T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 - 2.27T + 43T^{2} \) |
| 47 | \( 1 + 8.72T + 47T^{2} \) |
| 53 | \( 1 - 3.54T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 0.0862T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 3.50T + 71T^{2} \) |
| 73 | \( 1 - 7.22T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 18.6T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.506504710078796578637853863872, −7.57607315456393606440524126942, −6.91770421336266731627495061840, −6.41153957596125739276055364447, −5.43517855327760846390066246509, −4.72730160025746301839392563851, −3.63087522549451772960967624742, −3.03452156909777618959206625162, −2.52657973893336252805293740172, −0.880339991759257392340803712474,
0.880339991759257392340803712474, 2.52657973893336252805293740172, 3.03452156909777618959206625162, 3.63087522549451772960967624742, 4.72730160025746301839392563851, 5.43517855327760846390066246509, 6.41153957596125739276055364447, 6.91770421336266731627495061840, 7.57607315456393606440524126942, 8.506504710078796578637853863872
